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  "body": "\n## Summary\n\nThe forward difference operator $\\Delta_h$ sends a function $f$ to the function $x \\mapsto f(x + h) - f(x)$.\nIt is the discrete analogue of differentiation, and difference equations are often written with it.\nIt equals $E_h - I$, where $E_h$ is the [[shift operator|MBK6AE6734]] and $I$ is the identity.\n\n## Explanations\n\n### L1 human\n\nThe forward difference looks one step ahead and records how much the function changes.\nFor a sequence $u_0, u_1, u_2, \\dots$ with step $1$ it gives the sequence of gaps $u_1 - u_0, u_2 - u_1, \\dots$.\nFor the squares $0, 1, 4, 9, 16, \\dots$ the gaps are the odd numbers $1, 3, 5, 7, \\dots$.\nDoing it again gives the constant gaps $2, 2, 2, \\dots$, just as the second derivative of $x^2$ is the constant $2$.\n",
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